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This work considers two popular minimization problems: (i) the minimization of a general convex function $f(mathbf{X})$ with the domain being positive semi-definite matrices; (ii) the minimization of a general convex function $f(mathbf{X})$ regularized by the matrix nuclear norm $|mathbf{X}|_*$ with the domain being general matrices. Despite their optimal statistical performance in the literature, these two optimization problems have a high computational complexity even when solved using tailored fast convex solvers. To develop faster and more scalable algorithms, we follow the proposal of Burer and Monteiro to factor the low-rank variable $mathbf{X} = mathbf{U}mathbf{U}^top $ (for semi-definite matrices) or $mathbf{X}=mathbf{U}mathbf{V}^top $ (for general matrices) and also replace the nuclear norm $|mathbf{X}|_*$ with $(|mathbf{U}|_F^2+|mathbf{V}|_F^2)/2$. In spite of the non-convexity of the resulting factored formulations, we prove that each critical point either corresponds to the global optimum of the original convex problems or is a strict saddle where the Hessian matrix has a strictly negative eigenvalue. Such a nice geometric structure of the factored formulations allows many local search algorithms to find a global optimizer even with random initializations.
Matrix completion is about recovering a matrix from its partial revealed entries, and it can often be achieved by exploiting the inherent simplicity or low dimensional structure of the target matrix. For instance, a typical notion of matrix simplicit
We propose a new Riemannian geometry for fixed-rank matrices that is specifically tailored to the low-rank matrix completion problem. Exploiting the degree of freedom of a quotient space, we tune the metric on our search space to the particular least
Suppose that a solution $widetilde{mathbf{x}}$ to an underdetermined linear system $mathbf{b} = mathbf{A} mathbf{x}$ is given. $widetilde{mathbf{x}}$ is approximately sparse meaning that it has a few large components compared to other small entries.
Matrix sensing is the problem of reconstructing a low-rank matrix from a few linear measurements. In many applications such as collaborative filtering, the famous Netflix prize problem, and seismic data interpolation, there exists some prior informat
Goal: This paper deals with the problems that some EEG signals have no good sparse representation and single channel processing is not computationally efficient in compressed sensing of multi-channel EEG signals. Methods: An optimization model with L